Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, April 2015

A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada, and Director of the Centre for Mathematical and Statistical Sciences, India. He has published over 300 research papers and more than 25 books on topics in mathematics, statistics, physics, astrophysics, chemistry, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, President of the Mathematical Society of India, and a Member of the International Statistical Institute. He is the founder of the Canadian Journal of Statistics and the Statistical Society of Canada. He is instrumental in the implementation of the United Nations Basic Space Science Initiative. The paper is an attempt to capture the broad spectrum of scientific endeavors of Professor A.M. Mathai at the occasion of his anniversary.


Early Work in Design of Experiment and Related Problems
A.M. Mathai's first paper was in the area of Design of Experiment and Analysis of Variance in Statistics. This work was done after finishing M.A in Mathematics at University of Toronto and waiting to register for Ph.D, during July-August 1962. This was the first publication which appeared in the journal Biometrics in 1965, Mathai (1965) [Biometrics, 21(1965), [376][377][378][379][380][381][382][383][384][385]. This problem was suggested by Professor Ralph Wormleighton of the University of Toronto. In two-way classification with multiple observations per cell the analysis becomes complicated due to lack of orthogonality in the design. If two factors, such as the amount of fertilizer used and planting methods in an agricultural experiment to study the yield of corn, are to be tried and if the experiment is planned to replicate n times, it may happen that some observations in some replicate may get lost and as a result, instead of n observations per cell one may have n ij observations in the (i, j)th cell. When doing the analysis of the data, for estimating the effects of fertilizers, say, α 1 , ..., α p , one has to solve a singular system of linear equations of the type (I − A)α = G where G is known and I − A is singular and the unknown quantity α ′ = (α 1 , ..., α p ) is to be evaluated. Due to singularity, one cannot write α = (I − A) −1 G. This A = (a ij ) is the incidence matrix and has the property that all elements are positive and p j=1 a ij = 1 for each i = 1, ..., p. Mathai observed that this property means that a norm of A, namely A = max i p j=1 |a ij | = 1 and further, since the design is taking care of a general effect, one can impose a condition on α 1 , ..., α p such as α 1 + ... + α p = 0. Now, consider A being rewritten as A = A − C + C where C is a matrix where all the first row elements are equal to the median a 1 of the first row elements of A, all the second row elements are the median a 2 of the second row elements of A and so on. Now, by using the conditions on α i 's, Cα = O (null).
where B = (b ij ), b ij = a ij −a i , j = 1, ..., p or p j=1 |b ij | = p j=1 |a ij −a i | = sum of absolute deviations from the median a i , which is the least possible. Hence the norm B = max i p j=1 |a ij −a i | is the least possible and evidently < 1. Then works and others' related results were put together and brought out a monograph on characterizations, see A.M. Mathai and G. Pederzoli, Characterizations of the Normal Probability Law, Wiley Eastern, New Delhi and Wiley Halsted, New York, 1977.

Work in Multivariate Analysis
Mathai had already noted the densities of several structures could be written in terms of G and H-functions. Consider x 1 , x 2 , ..., x r , x r+1 , ..., x k mutually independently distributed positive random variables such as exponential variables, type-1 or type-2 beta variables or gamma variables or generalized gamma variables etc. Consider the structures where δ 1 , .., δ k are some arbitrary real powers. Then taking the Mellin transforms or the (s−1)th moments of u and v and then taking the inverse Mellin transform one can write the density of u as a G-function in most cases or as a H-function, and that of v as a H-function. Product of independently distributed type-1 beta random variables has the same structure of general moments of the likelihood ratio criterion or λ-criterion, or a one-to-one function of it, in many of testing hypotheses problems connected with one or more multivariate Gaussian populations and exponential populations. This showed that one could write the exact densities in the general cases as G-functions in most of the cases. Mathai was searching for computable representations in the general cases.

Development of 11-digit accurate percentage points for multivariate test statistics
Even after giving the explicit computable series forms for the various exact distributions of test statistics in the null (when the hypothesis is true) and non-null (under the alternate hypothesis) for the general parameters, the series forms were complicated and exact percentage points could not be computed. When Mathai visited University of Campinas in Brazil he met the physicist R.S. Katiyar. After six months of joint work of simplifying the complicated gamma products, psi and zeta functions, Katiyar was able to come up with a computer program. The first paper in the series giving the exact percentage points up to 11-digit accurate was produced. This paper made all the complicated theory usable in practical situations of testing of hypotheses in multivariate statistical analysis. The paper appeared in Biometrika and other papers followed, see Mathai and Katiyar (Biometrika, 66(1979), 353-356; Annals of the Institute of Statistical Mathematics, 31(1979), 215-224;Sankhya Series B, 42(1980), 333-341), Mathai (Journal of Statistical Computation and Simulation, 9(1979), 169-182).

Development of a computer algorithm for nonlinear least squares
After developing a computer program for computing exact 11-digit accurate percentage points from complicated series forms of the exact densities of λ-criteria for almost all multivariate test statistics, the problem of developing a computer program for non-linear least squares was re-examined. Starting with Marquardt's methods, there were a number of algorithms available in the literature but all these algorithms had deficiencies. There are a few (around 11) standard test problems to test the efficiency of a computer program. The efficiency of a computer program is measured by checking the following two items: In how many test functions the computer program fails and how many function evaluations are needed to come up to the final solution. These are the usual two criteria used in the field to test a new algorithm. A new algorithm for non-linear least squares was developed by Mathai and Katiyar which did not fail in any of the test functions and the number of function evaluations needed was least compared to all other algorithms available in the literature. The paper was published in a Russian journal, see Mathai and Katiyar (Researches in Mathematical Statistics (Russian), 207(10)(1993), 143-157). This paper was later translated into English by the American Mathematical Society.

Integer programming problem
The usual optimization problems such as optimizing a quadratic form or quadratic expression, subject to linear or quadratic constraints, optimizing a linear form subject to linear (linear programming problem) or quadratic constraints etc deal with continuous variables. When the variables are continuous then these optimization problems can be handled by using calculus or related techniques. Suppose that the variables can only take integer values such as positive integers 1, 2, 3, ... then the problem becomes complicated. Many of the standard results available when the variables are continuous are no longer true when the variables are integer-valued. One such problem was brought to the attention of Mathai by S. Kounias. This was solved and a joint paper was published, see Kounias and Mathai (Optimization, 19(1988), 123-131).

Work on Information Theory
When the exact distributions for the test statistics being worked out, side by side the work on information theory was also progressing. Characterizations of information and statistical concepts were the ones attempted as a joint venture by Mathai and Rathie. Several characterization theorems were established for various information measures and for statistical concepts such as covariance, variance, correlation etc, see for example, Mathai and Rathie (Sankhya Series A, 34(1972) where f (x) is a density function. This is the continuous version. There is also a discrete analogue. The denominator is put into the form of the exponent of 2 for ready applications to binary systems. When α → 1 one has H in (4.1) going to the Shannon entropy S = − ∞ −∞ f (x) ln f (x)dx and hence (4.1) is called an α-generalized entropy. There are several α-generalized entropies in the literature, including the one given by Mathai. This (4.1) in a modified form with the denominator replaced by 1 − α is developed later by C. Tsallis, as the basis for the whole area of non-extensive statistical mechanics. The Mathai-Rathie (1975) book can be considered to be the first book on characterizations. As a side result, as an application of functional equations, Mathai and Rathie solved a problem in graph theory, see Journal of Combinatorial Theory, 13(1972), 83-90. Other applications of information theory concepts in social sciences, population studies etc may be seen from Kaufman and Mathai (Journal of Multivariate Analysis, 3(1973), 236-242), Kaufman, Mathai, Rathie (Sankhya Series A(1972), 441-442), Mathai (Transactions of the 7th Prague Conference on Information Theory, pp. 353-357).

Applications to real-life problems
Applications of the concepts of information measures, 'entropy' or the measure of 'uncertainty', directed divergence (a concept of pseudo-distance), 'affinity' or closeness between populations, concept of 'distance between social groups' etc were applied to solve problems in social statistics, population studies etc. Mathai had developed a generalized measure of 'affinity' as well as 'distance between social groups'. On application side, dealing with applications of information theory type measures, see George and Mathai (Canadian Studies in Population, 2(1975), 91-100, 7(1980, 1-7; Journal of Biosocial Sciences (UK), 6(1975), 347-356; The Manpower Journal, 14(1978), 69-78).

Work on Biological Modeling
During one of the visits of Mathai to the Indian Statistical Institute in Calcutta, India, he came across the biologist T.A. Davis. Davis had a number of problems for which he needed answers. He had a huge collection of data on the number of petals in certain flowers of one species of plant. He noted that the petals were usually 4 in each flower but sometimes the number of petals was 5. He wanted to know whether the occurrence of 5-petaled flowers showed any pattern. His data were insufficient to come up with any pattern. Patterns, if any, would be connected to genetical factors. Then he had a question about how various patterns come in nature, in the growth of leaves, flowers, arrangements of petals and seeds in flowers etc and whether any mathematical theory could be developed to explain these. Then he brought in the observations on sunflower. When we look at flowers, certain flowers such as rose flower, sunflower etc look more beautiful than other flowers. This appeal is due to the arrangements of petals, florets, and color combinations. When we look at a sunflower at the florets or at the seed formations, after the florets dry up, we see some patterns in the arrangements of these seeds on the flower disk called capitulum. The seeds look like arranged along some spirals coming from the periphery going to the center. Let us call these as radial spirals. If one marks a point on the periphery and then one looks to the left of the mark one sees one set of radial spirals and if one looks to the right one sees a different set of radial spirals going in the opposite direction. The numbers of these two sets are always two successive numbers from a Fibonacci sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, ... (the sum of two successive numbers is the next number). Another observation made is that if one looks along a radial spiral this spiral does not go to the center but it becomes fuzzy after a while. At that stage if one draws a concentric circle and then look into the inside of this circle then one will see that if one started with the pair (13,21), then this has shifted to (8, 13) and then to (5, 8) and so on. The same sort of arrangement can be seen in pineapple, in the arrangement of leaves on a coconut tree crown and at many other places. If one takes a coconut crown and project onto a circle then the positions of the leaves on the crown form a replica of the seed arrangement in a sunflower. In a coconut crown if the oldest leaf is in a certain direction, call it 0-th direction then the next older leaf is not the next one to the oldest, but it is about θ degrees either to the right or to the left and this θ is such that θ 2π−θ = golden ratio = √ 5−1 2 . This golden ratio also appears at many places in nature and the above θ ≈ 137.5 o . Davis wanted a mathematical explanations for these and related observations. These observations were made by biologists over centuries. Many theories were also available on the subject. All the theories were trying to explain the appearance of radial spirals. Mathematicians try with differential equations and others from other fields try with their own tools. Mathai figured out that the radial spirals that one sees may be aftermath of something else and radial spirals are not generated per se. Also the philosophy is that nature must be working on very simple principles. If one buys sunflower seeds from a shop or look at sunflower seeds on a capitulum the seeds are all of the same dimensions if one takes one from the periphery or from any other spot on the capitulum. Such a growth can happen if something is growing along an Archimedes' spiral, which has the equation in polar coordinates r = kθ after one leaves the center. Davis' artist was asked to mark points on an Archimedes' spiral, differentiating from point to point at θ =≈ 137.5 o , something like a point moving along Archimedes' spiral at a constant speed so that when the first points reaches θ mark a second point starts, both move at the same speed whatever be the speed. When the second point comes to the mark θ a third point starts, and so on. After creating a certain number of points, may be 200 points, remove the Archimedes' spiral from the paper and fill up the space with any symmetrical object, such as circle, diamonds etc, with those points being the centers. Then if one looks from the periphery the two types of radial spirals can be seen. No such spirals are there but it is one's vision that is creating the radial spirals. Thus a sunflower pattern was recreated from this theory and Mathai and Davis proposed a theory of growth and forms. Consider a capillary a very thin tube with built-in chambers. Consider a viscous fluid being continuously pumped in from the bottom. The liquid enters the first chamber. When a certain pressure is built up, an in-built valve opens and the fluid moves into the second chamber and so on. Suppose that the tube opens in the center part of a pan (with a hole at the center). If the pan is fully sealed so that the only force acting on the liquid is Earth's gravity. The flow of the liquid will be governed by the functional equation f (θ 1 ) + f (θ 2 ) = f (θ 1 + θ 2 ) whose continuous solution is the linear function f (θ) = kθ. This is Archimedes' spiral.
The paper was sent for publication in the journal of Mathematical Biosciences the editor 'enthusiastically accepted for publication'. In this paper, Mathai and Davis * (Mathematical Biosciences, 20(1974), 117-133), a theory of growth and form is proposed. This theory still stands and since then there were many papers in physics, chemistry and other areas supporting various aspects of the theory and none has disputed the theory so far. In 1976 the journal has taken Mathai-Davis sunflower head as the cover design for the journal and it is still the cover design.

Work on coconut tree crown
The coconut crown was also examined from many mathematical points of view and found to be an ideal crown. This paper may be seen from Mathai and Davis (Proceedings of the National Academy of Sciences, India, 39(1973), 160-169).

Engineering wonder of Bayya bird's nest and other biological problems
Further problems looked into by Mathai and Davis are the following: (1) The engineering aspect of the egg chamber of bayya bird's nest. The nest hangs from the tips of tree branches, the mother bird goes into the egg chamber through the tail opening of the nest, the nest oscillates violently during heavy winds or storms but no egg comes out of the egg chamber and fall through the tail opening. Naturally the tail opening is bigger than the diameter of the eggs because the mother bird goes through that opening. This shape, beng an engineering marvel, was examined by Mathai and Davis. (2) thermometer birds in Andaman Nicobar Islands; (3) transfer of Canadian Maple Syrup technology in the production of palm sugar and jaggery in Tamilnadu, India; (4) Nipa palms to prevent sea erosion along Kannyakumari sea coast; (5) rejuvenation of Western Ghats in Kannyakumari region. All these projects were undertaken jointly by the Centre for Mathematical Sciences, Trivandrum Campus (CMS) where A.M. Mathai was the Honorary Director and Haldane Research Institute of Nagarcoil, Tamilnadu (HRI) where T.A. Davis was the Director and A.M. Mathai was the Honorary Chairman. Earlier to these studies, George and Mathai had done work in population problems, especially in the study of inter-live-birth intervals, that is, the interval between two live births among women in child-bearing age group, see George and Mathai (Sankhya Series B, 37(1975), 332-342;Demography of India, 5(1976), 163-180;The Manpower Journal, 14(1978), 69-78). Here, Mathai had introduced the concepts of affinity and distance between social groups.

Introducing the phrase 'statistical sciences'
By 1970 Mathai was working to establish a Canadian statistical society and a Canadian journal of statistics. The phrase 'statistical sciences' was framed and defined it as a systematic and scientific study of random phenomena so that the theoretical developments of probability and statistics and applications in all branches of knowledge will come under the heading 'statistical sciences', and random variables as an extension of mathematical variables or mathematical variables as degenerate random variables. After launching Statistical Science Association of Canada, the term 'statistical science' became a standard phrase. Journals and organizations started using the name 'statistical science'. Mathai was responsible to introduce these terms into scientific literature.
When G.P.H. Styan, a colleague of Mathai, was editing the news bulletin of the Institute of Mathematical Statistics he posed the question whether the phrase 'statistical science' was ever used before launching statistical science association of Canada. There was a response from a Japanese scientist claiming that he had used the term 'statistical science' before. Incidentally, later the Institute of Mathematical Statistics changed the name of Annals of Mathematical Statistics to Annals of Statistics and hence that name was no longer available when statistical science association of Canada changed its name back to the original proposed name Statistical Society of Canada.

Work on Probability and Geometrical Probabilities
Work in mathematical statistics and special functions continued. As a continuation of the investigation of structural properties of densities, Mathai came across the distributions of lengths, areas and volume contents of random geometrical configurations such as random distance, random area, random volume and random hyper-volume. All the theories of G and H-functions, products and ratios of positive random variables etc could be used in examining the distributional aspects of volume of random parallelotopes and simplices. By analyzing the structure of general moments, Mathai noted that these could be generated by products of independently (1) gamma distributed points, (2) uniformly distributed points, (3) type-1 beta distributed points, (4) type-2 beta distributed points. Out of these, (1) fell into the category of G p,0 0,p , the second and third fell into G p,0 p,p category and (4) fell into G p,p p,p category, for all of which the necessary theory was already developed by Mathai and his team. Papers were published on the distributional aspects, see Mathai (Sankhya Series A. 45(1983), 313-323; Mathai and Tracy (Communications in Statistics A,12 (15)(1983) (2000), 219-232), Mathai and Pederzoli (American Journal of Mathematical and Management Sciences 9(1989), 113-139;Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl., 50(1997), 235-258).

A conjecture in geometric probabilities
Then Mathai came across a conjecture posed by an Australian scientist R.E. Miles, regarding the asymptotic normality of a certain random volume coming from uniformly distributed random points. This was proved to be true by H. Ruben. In fact Ruben brought this area to the attention of Mathai. The structure of the random geometric configuration was known to Mathai and that it was a G-function of the type G p,0 p,p and Mathai realized that a very simple proof of the conjecture could be given by using the asymptotic formula, or Stirling's formula which is the first approximation there, for gamma functions. This was worked out and shown that the conjecture could be proved very easily. This paper appeared in the journal in probability, see Mathai (Annals of Probability. 10(1982), 247-251). Incidentally, there is a mistake there. Final representation is given in terms of a confluent hypergeometric function 1 F 1 there but it should be a Gauss hypergeometric function 2 F 1 , one parameter is missed there in writing the final form. Then Mathai noted that the same conjecture can be formulated in terms of type-1 beta distributed random points and similar conjectures could be formulated for type-2 beta distributed random points and gamma distributed random points. These conjectures were formulated and solved, see Mathai (Sankhya Series A, 45(1983), 313-323; American Journal of Mathematical and Management Sciences, 9(1989), 113-139); Mathai and Tracy (Communications in Statistics A, 12(15)(1983), 1727-1736Metron, 44(1986), 101-110).

Random volumes and Jacobians of matrix transformations
Side by side Mathai was developing functions of matrix argument. The work in this area will be given later but its connection to geometrical probabilities will be mentioned here. The area of stochastic geometry or geometrical probabilities is a fusion of geometry and measure theory. When measure theory is mixed with geometry the standard axiomatic definition for probability measure is not sufficient. It is quite evident to see that an additional property of invariance is needed because a geometrical object can be moved around in a plane or in space and the probability statements must remain the same. The famous Betrand's paradoxes or Russell's paradoxes come from lack of invariance conditions there. The details are discussed in the book, A.M. Mathai,Introduction to Geometrical Probability: Distributional Aspects and Applications, Gordon and Breach, New York, 1999. Consider a circle of radius r. Take two points A and B at random and independently on the circumference of this circle. Here, 'at random' could mean that the probability of finding a point, such as A, in an interval of length δ is δ 2πr . Consider the chord AB. Then AB is a random chord. Let P be the mid point of this chord and O the center of the circle. Then OP is fixed when AB is fixed and OP is perpendicular to AB. Consider another situation of selecting a point P at random inside the circle. This can be done by assigning probability of finding P in a region R inside the circle is R πr 2 . If P is fixed and if P is the midpoint of a chord then the chord is automatically fixed. In many ways one can geometrically uniquely determine a chord. The chord can be made 'random' by assigning probabilities in many ways. Two ways are described above. If one asks a question, what is the probability that the length of this random chord is less than a specified number? The answer will be different for different ways of assigning probabilities. This is the paradox. Note that all steps in the derivations of the answers will be correct and valid steps as per the usual axioms of probability.
In stochastic probability area the methods used are the methods from differential and integral geometry and usually very difficult. Even if one wishes to talk about the distribution of random volume of a parallelotope through differential or integral geometry the process is very involved. Mathai noted that such problems could be easily answered through Jacobians of matrix transformations. A paper was published in advances in applied probability, see Mathai Advances in Applied Probability, 31 (2)(1999)

Applications in transportation problems
As an application of geometrical probability problems Mathai explored the travel distance from the suburb to city core for circular and rectangular grid cities. Many of the European cities are designed with a city center and circular and radial streets from the center whereas in North America most of the cities are designed in rectangular grids. Travel distances, time taken and associated expenses are random quantities and related to the nature of city design. Some problems of this type were analyzed by Mathai (Environmetrics, 9(1998), 617-628); Mathai and Moschopoulos (Environmetrics, 10(1999), 791-802).

Work in Astrophysics
After publishing the two books on generalized hypergeometric functions in 1973 and Hfunction in 1978, physicists were interested to use those results in their works. A number of people from different parts of Germany were using these results. The German group working in astrophysics problems were trying to solve some problems connected with reaction rate theory. Then H.J. Haubold, came to McGill University with open problems where help from special function theory was needed. After converting their problems in terms of integral equations, Mathai noted that the basic integral to be evaluated was of the following form: x γ e −ax−bx − 1 2 dx (7.1) and generalizations of this integral. Note that if a or b is zero then the integral can be evaluated by using a gamma integral. Mathematically, if the nonlinear exponent is of the form x − 1 2 or of the form x −ρ , ρ > 0 it would not make any difference. Mathai could not find any such integrals in any of the books of tables of integrals. He noted that the integrand consisted of integrable functions and therefore one could make statistical densities out of them. For example, f 1 (x) = c 1 x γ e −ax , 0 ≤ x < ∞ is a density where c 1 is the normalizing constant. Similarly f 2 (x) = c 2 e −x ρ , ρ > 0, 0 ≤ x < ∞ is a density where c 2 is the normalizing constant. Then the structure in (7.1) can be written as follows: where g(u) can represent the density of u = x 1 x 2 where x 1 and x 2 are independently distributed positive real scalar random variables with the densities f 1 (x 1 ) and f 2 (x 2 ) respectively. Once the structure in (7.1) is identified as that in (7.2) then, since the density being unique, it is only a matter of finding the density g(u) by using some other means. We can easily use the properties of arbitrary moments. For example due to statistical independence of x 1 and x 2 , where E denotes the expected value. Note that E(x s−1 1 ) is available from f 1 (x 1 ) and E(x s−1 2 ) from f 2 (x 2 ). Then g(u) is available from the inverse, that is, where i = √ −1 and c is determined from the poles of E(u s−1 ). Thus, by using statistical techniques the integral in (7.1) was evaluated. After working out many results it was realized that one could also use Mellin convolution of a product to solve integrals of the type in (7.1). This was not seen when the method through statistical distribution theory was devised. Various types of thermonuclear reactions, resonant, non-resonant, depleted case, high energy cut off case etc were investigated. The work also went into exploring exact analytic solar models, gravitational instability problems, solar neutrino problems, reaction-rates, nuclear energy generation etc. The work until 1988 was summarized in the monograph Mathai and Haubold (Modern Problems in Nuclear and Neutrino Astrophysics, Akademie-Verlag, Berlin, 1988). Since then a lot of work was done, some of them are the following: Haubold and Mathai (Annalen der Physik, 44(1987), 103-116; Astronomische Nachrichten, 308 (5)(1987)

Work on Differential Equations
One of the problems investigated in connection with problems in astrophysics was the gravitational instability problem. The problem was brought to the attention of Mathai by Haubold. Papers by Russian researchers were there on the problem of mixing two types of cosmic dusts. Mathai looked at it and found that by making a transformation in the dependent variable and by changing the operator to t d dt instead of the integer order differential operator D = d dt one could identify the differential equation as a particular case of the differential equation satisfied by a G-function. Then G-function theory could be used to solve the problem of mixing k different cosmic dusts. Thus the first paper in integer order differential equation was written and published in the MIT journal, see Mathai (Studies in Applied Mathematics, 80(1989), 75-93). Two follow-up papers were written developing the differential equation and applying to physics problems, see Haubold and Mathai (Astronomische Nachrichten,312(1)(1991), 1-6; Astrophysics and Space Science, 214(1&2)(1994), 139-149).

The Idea of Laplacianness of Bilinear Forms and Work on Quadratic and Bilinear Forms
In the 1980's two students of Mathai, S.B. Provost and D. Morin-Wahhab, finished their Ph.Ds in the area of quadratic form. Mathai has also published a number of papers on quadratic and bilinear forms by this time. Then it was decided to bring out a book on quadratic forms in random variables. On the mathematical side, there were books on quadratic forms but there was none in the area of quadratic forms in random variables. Only real random variables and samples coming from Gaussian population were considered. Later in 2005 Mathai extended the theory to cover very general classes of populations. This aspect will be considered later when pathway models are discussed. Only when I. Olkin pointed out to Mathai about the many applications of complex Gaussian case in communication theory, after the book appeared in print, Mathai and Provost realized that an equal amount of material was missed out: A.M. Mathai and S.B. Provost, Quadratic Forms in Random Variables: Theory and Applications, Marcel Dekker, New York, 1992. Work on quadratic forms and related topics may be seen from Mathai(Communications in Statistics A,20 (10)

Chisquaredness of quadratic forms and Laplacianness of bilinear forms
Consider the following quadratic form and bilinear form: where x 1 , ..., x p , y 1 , ..., y q are real scalar random variables, A is a p × p matrix and B is a p × q matrix, where p ≤ q or p ≥ q. When X is distributed as a N p (O, I) or a p-variate Gaussian or normal population with mean value null and covariance matrix an identity matrix then there is a theorem which says that Q is distributed as a chisquare with r degrees of freedom if and only if A is idempotent and of rank r. This result is frequently used, especially in design of experiments and analysis of variance problems. In fact, this result and its companion result on the independence of two quadratic forms are the backbones of the areas of analysis of variance, analysis of covariance, regression, model building and many others. What is a concept corresponding to chisquaredness of quadratic form in the bilinear form case? It was shown by Mathai that the concept is Laplacianness or the corresponding distribution is Laplace density instead of chisquare density, see Mathai (Journal of Multivariate Analysis, 45(1993), 239-246). Apart from introducing the concept of Laplacianness, this paper also throws light on covariance structures. When Mathai was taking his M.A. degree in mathematics, one of the professors in a course on multivariate analysis asked a simple-looking question in 1962. If one has a simple random sample from a bivariate real normal population N 2 (µ, Σ), Σ > 0 (positive definite; standard notation), consider the sample correlation coefficient, denoted by r, where The question was what is the density of the sample covariance n j=1 (x ij −x)(y ij −ȳ)/n? The density of r in the bivariate normal case, and the corresponding density for the sample multiple correlation in the multivariate case, were already available in the literature. The answer looked trivial because the sample covariance is directly connected to the sample correlation. Nobody had the answer including the professor who posed the question. In 1990-19991 when Mathai was writing on Laplacianness, he realized that covariance structure is nothing but a bilinear form and hence the density of the sample covariance must be available from that of the bilinear form. Thus, the 1962 question was answered in the above-mentioned 1993 paper. The corresponding matrix-variate case should also be available but nobody has worked out yet.

Bilinear form book
After publishing the quadratic form book in 1992, a lot of work had been done on bilinear forms. Even though a bilinear form can be written as a quadratic form, there are many properties enjoyed by bilinear form and not enjoyed by quadratic forms. Quadratic forms do not have covariance structures. Then T. Hayakawa of Japan contacted Mathai asking why not bring out a book on bilinear form, parallel to the one on quadratic form including chapters on zonal polynomials. This book on bilinear forms and zonal polynomials was brought out in 1995: A.M. Mathai, S.B. Provost and T. Hayakawa, Bilinear Forms and Zonal Polynomials, Springer, New York, 1995, in the lecture notes series. Additional papers may be seen from Mathai and Pederzoli (Journal of the Indian Statistical Society, 3(1995), 345-356; Statistica, LVI(4)(1996), 4-7-41).

Functions of Matrix Argument
Meanwhile Mathai's work on functions of matrix argument was progressing. These are realvalued scalar functions where the argument is a real or complex matrix. The theory is well developed when the argument matrix is real positive definite or hermitian positive definite. Note that when A is a square or rectangular matrix we do not have a concept corresponding to the square root of a scalar quantity uniquely defined. But if the matrix A is real positive definite or hermitian positive definite, written as A > O, operations such as square root can be uniquely defined. Hence the theory is developed basically for real positive definite or hermitian positive definite matrices. Gordon and Mathai tried to develop a matrix series and a pseudo analytic function involving general matrices, the attempt was not fully successful but some characterization theorems for multivariate normal population could be established, see Gordon and Mathai (Annals of Mathematical Statistics, 43(1972), 205-229). Gordon has two more papers in the area, one in the Annals of Statistics and the other in the Annals of the Institute of Statistical mathematics. Hence the theory of real-valued scalar functions of matrix argument is developed when the matrix is real or hermitian positive definite. There are three approaches available in the literature. One is through matrix-variate Laplace transform and inverse Laplace transform developed by C. Herz and others, see for example, Herz (Annals of Mathematics,61(3)(1955), 474-523). Here one basic assumption is functional commutativity f (AB) = f (BA) even if AB = BA, where A and B are p × p matrices. Under functional commutativity we have the following result, observing that when A is symmetric there exists and orthonormal matrix P, P P ′ = I, P ′ P = I such that P ′ AP = D where D is a diagonal matrix with the diagonal elements being the eigenvalue of A. Then Thus, the original function of p(p + 1)/2 real scalar variables, can be reduced to a function of p variables, the eigenvalues of A. Another approach is through zonal polynomials, developed by Constantine, James and others, see for example James (Annals of Mathematics, 74(1961), 456-469) and Constantine (Annals of Mathematical Statistics, 34(1963Statistics, 34( ), 1270Statistics, 34( -1285. In this definition a general hypergeometric function with r upper parameters and s lower parameters is defined as follows: where C K (X) is zonal polynomial of order k, K = (k 1 , ..., k p ), k 1 + ... + k p = k, and for example, through the definition of Laplace and inverse Laplace pair. The third approach is due to Mathai and it is defined in terms of a general matrix transform or M-transform. The M-transform of f (−X) defined by the equation where ℜ(·) means the real part of (·). Under functional commutativity, f (−X) in ( In this definition, a general hypergeometric function with r upper and s lower parameters will be defined as that class of functions for which the M-transform is the following: where Γ p (a) is the real matrix-variate gamma given by Then that class of function f (−X) is given by the equation ( Series II, Suppl., 65(2000), 219-232; Linear Algebra and Its Applications, 183(1993), 202-221;in Probability and Statistical Methods with Applications, pp.293-316, Chapman and Hall, 2001), Mathai and Saxena (Journal de Matematica e Estatistica, 1(1979), 91-106), Mathai and Rathie (Statistica, XL(1980), 93-99;Sankhya Series A, 42(1980), 78-87;), Mathai and Tracy (Communications in Statistics A, 12(15)(1983), 1727-1736Metron, 44(1986), 11-110), Mathai and Pederzoli (Metron, LI(3-4)(1993), 3-24;Indian Journal of Pure Applied Mathematics, 27(3)(1996), 7-32;Linear Algebra andIts Applications, 253(1997), 209-226, 269(1998), 91-103). The important publication in this area is the book on Jacobians of matrix transformation: A.M. Mathai, Jacobians of Matrix Transformations and Functions of Matrix Argument, World Scientific Publishing, New York, 1997. The work on functions of matrix argument is continuing in the form of applications in pathway models, fractional calculus and so on. These will be mentioned later.
In connection with matrix-variate integrals it is a very often asked question that whether matrix-variate integrals can be evaluated by treating them as multiple integrals and by using standard techniques in calculus. Mathai explored the possibility of explicitly evaluating matrixvariate gamma and beta integrals as multiple integrals in calculus. The basic matrix-variate integrals are the gamma integral and beta integrals, where X is a p × p real positive definite matrix or hermitian positive definite matrix. For example, when X is real and X > O (positive definite) the gamma integral is The corresponding integrals are there in the complex-variate case also. It is shown that this can be done explicitly for p = 2 and a recurrence relation can be obtained so that step by step they can be evaluated but for p > 2 this method of treating as multiple integrals is not a feasible

Power transformation and exponentiation
Another problem explored is to see the nature of models available by power transformations and exponentiation of standard probability models. Such a study is useful when looking for an appropriate model for a given data. These explorations are done in Mathai (Journal of the Society for Probability and Statistics (ISPS), 13(2012), 1-19).

Symmetric and asymmetric models
A symmetric model, symmetric at x = a where a could be zero also, means that for x < a the behavior of the function or the shape of the function is the same as its behavior for x > a. In many practical situations, symmetry may not be there. The behavior for x < a may be different from that for x > a. Many authors have considered asymmetric models where asymmetry is introduced by giving different weighting factors for x < a and for x > a so that the total probability under the curve will be 1. But the shape of the curve itself may change for x < a and for x > a. A method is proposed in the paper referred to in 11.1 above (Mathai 2012) where asymmetry is introduced through a scaling parameter so that the shape itself will be different for x < a and x > a cases but the total probability remaining as 1, which may have more practical relevance.

The Pathway Model
The basic idea was there in a paper of 1970's in the area of population studies where it was shown that by a limiting process one can go from one class of functions to another class of functions, the property is basically coming from the theory of hypergeometric functions from the aspect of getting rid off a numerator or a denominator parameter. This idea was revived and written as a paper on functions of matrix argument where the variable matrix is a rectangular one, see Mathai (Linear Algebra and Its Applications, 396(2005), 317-328). Let X be a real m × n matrix, m ≤ n and of rank m be a matrix variable. Let A be m × m and B be n × n constant nonsingular matrices. Consider the function where α, η, C be scalar constants. This C can act as a normalizing constant if we wish to create statistical density out of (12.1). Consider the case when m = 1, n = 1 and x > 0. Then one can also take powers for x and the model in (12.1) can be written as where a > 0, δ > 0, η > 0, x ≥ 0. In the matrix-variate case in (12.1) arbitrary powers for matrices is not feasible even though AXBX ′ is positive definite because even for a positive definite matrix, Y , arbitrary power such as Y δ may not be uniquely defined. Even when uniquely defined transformation such as Z = Y δ will create problems when computing the Jacobians. The types of difficulties that can arise may be seen for the case δ = 2 described in the book, A.M. Mathai, Jacobians of Matrix Transformations and Functions of Matrix Argument, World Scientific Publishing, New York 1997. Hence for the matrix case we consider only when δ = 1. Consider case −∞ < α < 1. Then (12.2) remains as it is given in (12.2) which is a generalized type-1 beta function. But if α > 1 then writing 1 − α = −(α − 1) the form in (12.2) changes to the following: for a > 0, α > 1, η > 0, δ > 0, x ≥ 0. This model is a generalized type-2 beta model. When α → 1 in (12.2) and (12.3), f 1 (x) and f 2 (x) reduce to the the form This is a generalized gamma model. Thus three functional forms f 1 (x), f 2 (x), f 3 (x) are available for α < 1.α > 1, α → 1. This parameter α is called the pathway parameter, a pathway showing three different families of functions.
The practical utility of the model is that if (12.4) is the stable or ideal situation in a physical system then the unstable neighborhoods or functions leading to (12.4) are given in (12.2) and (12.3). In a model building situation, if the underlying data show a gamma-type behavior then a best-fitting model can be constructed for some values of the parameters or for some value of α the ideal model can be determined. Most of the statistical models in practical use in the areas of statistics, physics and engineering fields can be seen to be a member or products of members from f 1 , f 2 , f 3 above. Note that for α > 1 and α → 1 situations we can take δ > 0 or δ < 0 and both these situations can create statistical densities. Note that f 1 is a family of finite range models whereas f 2 and f 3 are families of infinite range models. Extended models are available by replacing x by |x| so that the whole real line will be covered. In this case the nonzero part of model (12.2) will be in the range ±[a(1 − α)] − 1 δ and for others −∞ < x < ∞. Note that in (12.1) all individual variables x ij 's are allowed to vary over the whole real line subject to the condition I − (1 − α)AXBX ′ > O (positive definite). This model is also extended to complex rectangular matrix-variate case, see Mathai and Provost (Linear Algebra and Its Applications, 410(2005), 198-216).
Note that (12.2) for γ = 0, δ = 1, a = 1, η = 1 is Tsallis statistics in nonextensive statistical mechanics. The function, without the normalizing constant c 1 will then be g(x) = [1 − (1 − α)x] 1 1−α (12.5) which is Tsallis statistics. This can be generated by optimizing Tsallis entropy or Havrda-Charvát entropy with the denominator factor 1 − α instead of 2 1−α − 1, subject the constraint that the first moment is fixed and this condition can be connected to the principle of the total energy being conserved. Note that (12.5) is also a power function model.
Mathai's students have introduced a pathway fractional integral operator based on (12.2) and a pathway transform based on (12.2) and (12.3). (12.2),(12.3) can also be obtained by optimizing Mathai's entropy subject to two moment type constraints and also the pathway parameter α can be derived in terms of moments of f 1 (x) or f 2 (x). Thus, in terms of entropies one can establish a entropic pathway, in terms of distributions as explained above one can create a distributional pathway, one can also look into the corresponding differential equations and create a differential pathway, in current use are the following: There is no condition on the parameter γ. If these are to be written in terms of H-functions then α and γ have to be real and positive. A generalization can be made by introducing a general hypergeometric type function, which may be written as E a1,...,ar α,β,b1,...,bs (x δ ) = ∞ k=0 (a 1 ) k ...(a r ) k (x δ ) k k!Γ(β + αk)(b 1 ) k ...(b s ) k where the notation (a j ) k and (b j ) k are Pochhammer symbols. Convergence conditions can be worked out for this general form.
A problem of interest in this case is a general Mittag-Leffler density because such a density is needed in non-Gaussian stochastic processes and time series areas. Such a density was introduced based on E γ α,β (x δ ) and it is shown that such a model is connected to fat-tailed models, Lévy, Linnik models. Structural properties and asymptotic behavior are also studied and it is shown that such models are not attracted to Gaussian models, see Mathai (Fractional Calculus & Applied Analysis,13(1) (2010), 113-132), Mathai and Haubold (Integral Transforms and Special Functions, 21(11)(2011), 867-875).

Work on Krätzel Function and Krätzel Densities
Another area explored is Krätzel function, Krätzel transform and Krätzel densities. Since Krätzel transform is important in applied analysis area, a general density is introduced based on Krätzel integral. The basic Krätzel integral is of the form x γ e −ax− y x dx, a > 0, y > 0 (14.1) which can be generalized to the form for a > 0, y > 0, α > 0, β > 0 or β < 0. The integrand in (14.1), normalized, is the inverse Gaussian density. The integral itself can be interpreted as Mellin convolution of a product, the marginal density in a bivariate case etc. The integral in (14.2) is connected the general reaction-rate probability integral in reaction-rate theory (β = 1 2 , α = 1 is the basic integral in reaction-rate theory) , unconditional densities in Bayesian analysis, marginal densities in a